The original argument, due to Lieb, Schultz, and Mattis (LSM) [1]}, applied to an \(N\) -site, spin-\(S\)  \(XXZ\)  spin chain with periodic boundary conditions and zero total magnetization. LSM showed that if the ground state \({| \, \Psi ^{\vphantom{\dagger }}_0 \, \rangle }\)  is a state with crystal momentum \(K^{\vphantom{\dagger }}_0\) , i.e. if \({\hat{t}}\,{| \, \Psi ^{\vphantom{\dagger }}_0 \, \rangle }=e^{iK^{\vphantom{\dagger }}_0}{| \, \Psi ^{\vphantom{\dagger }}_0 \, \rangle }\) , where \({\hat{t}}\)  is the lattice translation operator and we work in units where the lattice spacing is \(a\equiv 1\) , then the application of the spin twist operator,
\(\hat{V}=\exp \!\bigg ({{2\pi i}\over N}\sum _{j=1}^N j\,{\hat{S}}^z_j\bigg )\quad ,\) 
